Various tiling systems are known for creating a mosaic, or tessellation, for covering walls, floors, ceilings, streets or paths and also for producing toys, games and various structures. Usually these tiles are formed from simple polygons such as triangles, squares, rectangles, and octagons, which results in a plane-filling pattern that repeats, i.e, is periodic. These systems, while functional and easy to install, result in a somewhat boring and predictable pattern.
In addition, other tiling systems are known which do not use simple polygons; however, many of these also provide a periodic pattern and some of these are incapable of completely filling a plane, i.e., there are gaps in between various sets.
Examples of these prior systems are disclosed in the following U.S. Pat. Nos.: 3,921,312 to Fuller; 3,981,505 to Odier; 4,133,152 to Penrose; 4,223,890 to Schoen; 4,343,471 to Calvert; and 4,350,341 to Wallace. A further example of such a system is disclosed in New Mathematical Pastimes by MacMahon, 1921, Cambridge at the University Press, pages 50-59.